# MST124 BOOK A UNIT 1

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## 3

Strategy
To find the positive factor pairs of a positive integer
- try dividing the integer by each of the numbers 1, 2, 3, 4,... in turn
- whenever you find a factor, write it down along with the other factor in the factor pair
- stop when you get a factor pair that you have already

The fundamental theorem of arithmetic
Every integer greater than 1 can be written as a product of prime factors in just one way
- except that you can change the order of the factors

Strategy
To find the prime factorisation of an integer greater than 1
- repeatedly 'factor out' the prime 2 until you obtain a number that isn't divisible by two
- repeatedly factor out the prime 3 until you obtain a number that isn't divisible by three
- repeatedly factor out the prime 5 until you obtain a number that isn't divisible by five
- continue this process with each prime number in turn, and stop when you have a produce of primes
- write out the prime factorisation with the numbers in increasing order, using index notation

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## 4

Strategy
To find the lowest common multiple or highest common factor of two or more integers greater than 1
- find the prime factorisations of the numbers
- to find the LCM, multiply together the highest power of each prime factor occurring in any of the numbers
- the find the HCF, multiply together the lowest power of each prime factor common to all the numbers
(page 23 book A)

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## 5

Strategy
To simplify a term
- find the overall sign and write it at the front
- simplify the rest of the coefficient and write it next
- write any remaining parts of the term in an appropriate order

Strategy
To simplify an expression with more than one term
- identify the terms
- each term after the first starts with a plus or minus sign that isn't inside brackets
- simplify each term, including the sign at the start of each term
- collect any like terms

Strategy
To multiply out brackets
- multiply each term inside the brackets by the multiplier

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## 6

Strategy
- if the sign is plus, keep the sign of each term inside the brackets the same
- if the sign is minus, change the sign of each term inside the brackets

- for any expressions A and B, the negative of A-B is B-A

- multiplying an expression that contains several terms by a second expression is the same as multiplying each term of the first expression by the second expression
- dividing an expression that contains several terms by a second expression is the same as dividing each term of the first expression by the second expression

Strategy
To multiply out brackets in an expression with more than one term
- identify the terms
- multiply out the brackets in each term, including the sign at the start of each resulting term
- collect any like terms

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## 7

Difference of two squares
- for any expressions A and B
(A+B)(A-B) = A2 - B2

(A+B)2 is not equal to A2 + B

Squaring brackets
(A+B)2 = A2 + 2AB + B2
(A-B)2 = A2 - 2AB + B2

Strategy
To take out a common factor from an expression
- Find a common factor of the terms (usually the HCF)
- write the common factor in front of a pair of brackets
- write what is left of each term inside the brackets

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## 8

Strategy
To add or subtract algebraic fractions
- make sure that the fractions have the same denominator
- if necessary, rewrite each fraction as an equivalent fraction to achieve this
- add or subtract the numerators
- simplify if possible

Strategy
To multiply or divide algebraic fractions
- to multiply two or more algebraic fractions, multiply the numerators together and multiply the denominators together
- to divide by an algebraic fraction, multiply by its reciprocal
- simplify if possible

- for every even natural number n, every positive number has exactly two real nth roots
- one positive one negative
- every negative number has no real nth roots
- the number 0 has 1 nth root - itself
- for every odd natural number n, every real number has exactly one real nth root

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## 9

- the square root of a product of numbers is the same as the product of the square roots of the numbers
- the same rule applies to quotients
√ab = √a√b
√(a/b) = √a/√b

- you can rationalize the denominator of the surd a/√b by multiplying the top and bottom by √b

- when the denominator is a sum of two terms, either or both of which is a rational number multiplies by an irrational square root, rationalise the denominator by multiplying the top and bottom of the fraction by a conjugate of the expression in the denominator
- this is an expression that is obtained by changing the sign of one of the two terms (generally the second term)

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## 10

Index laws for a single base
-to multiply two powers with the same base, add the indices
aman = am+n
- to divide two powers with the same base, subtract the indices
am/an = am-n
- to find a power of a power, multiply the indices
(am)n = amn
- a number raised to the power 0 is 1
a= 1
- a number raised to a negative power is the reciprocal of the number raised to the corresponding positive power
a-n = 1/an

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## 11

Index laws for two bases
- a power of a product of numbers is the same as the product of the same powers of the numbers
- this rule applies for a power of a quotients
(ab)n = anbn
(a/b)n = an/bn

a-n = 1/an
1/a-n = an

Converting between fractional indices and roots
a1/nn√a
am/n = (n√a)mn√(am)

Scientific Notation
To express a number in scientific notation, write it in the form:
(a number between 1 and 10, but not including 10) x (an integer power of ten)
e.g., 427 = 4.27 x 100 = 4.27 x 102

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## 12

Rearranging equations
Carrying out any of the following operations on an equation gives an equivalent equation
- rearrange the expressions on one or both sides
- swap the sides
- do the same thing to both sides

Doing the same thing to both sides of an equation
Doing any of the following things to both sides of an equation gives an equivalent equation
- subtract something
- multiply by something (provided that it is non-zero)
- divide by something (provided that it is non-zero)
- raise to a power (provided that the power is non-zero, and that the expression on each side of the equation can take only non-negative values)

- moving a term of one side of an equation to the other side, and changing its sign gives an equivalent equation
(change the side, change the sign)

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## 13

- multiplying each term on both sides of an equation by something, provided that it is non-zero, gives an equivalent equation
- dividing each term on both sides of an equation by something, once again providing its non-zero, gives an equivalent equation

Cross Multiplying
If A, B, C, and D are any expressions, then the equations:
(A/B) = (C/D) and AD = BC
are equivalent (provided that B and D are never zero)

Strategy
To solve a linear equation in one unknown
- clear any fractions and multiply out any brackets
- to clear fractions, multiply through by a suitable expression
- add or subtract terms on both sides to get all the terms in the unknown on one side, and all the other terms on the other side
- collect like terms
- divide both sides by the coefficient of the unknown

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## 14

Strategy
To make a variable the subject of an equation (this works for some equations but not all)
- clear any fractions and multiply out any brackets
- to clear fractions, multiply through by a suitable expression
- add or subtract terms on both sides to get all the terms containing the required subject on one side, and all the other terms on the other side
- collect like terms
- if more than one term contains the required subject, then take it out as a common factor
- divide both sides by the expression that multiplies the required subject

- this strategy works provided that the equation is 'linear in the required subject'
- its form must be such that if you replace every variable other than the required subject but a suitable number (one that doesn't lead to multiplication or division by 0) then the result is a linear equation in the required subject

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